
%!TEX program = xelatex
%!TEX TS-program = xelatex
%!TEX encoding = UTF-8 Unicode

\documentclass[10pt]{article} 

\input{wang_preamble.tex}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%页面的高度与宽度、上边界与左边界-------正常打印边界
%\addtolength{\textheight}{1cm}
%\addtolength{\voffset}{-1cm}
\addtolength{\textwidth}{1.5cm}
\addtolength{\hoffset}{-1cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{titling}
\setlength{\droptitle}{-2cm}   % This is your set screw

%%文档的题目、作者与日期

%\author{王立庆（2019级数学与应用数学1班）}
\author{学号 \underline{\hspace{4cm}} 姓名  \underline{\hspace{4cm}} }
%\title{高等代数第六章：向量空间}
\title{第八章欧氏空间（8.3-8.4）考试 }
%\date{\vspace{-3ex}}
\renewcommand{\today}{\number\year \,年 \number\month \,月 \number\day \,日}
\date{2023年5月17日}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\maketitle

\thispagestyle{empty}

%\begin{abstract}
%%主要内容：
%7.3. 
%7.4. 
%7.5. 

%
%\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %1
设 $V=\mathbb{R}^2$ 是平面，设 $\sigma: V\to V$ 是关于直线 $y=2x$ 的反射。
\begin{enumerate}
\item  设 $\alpha=\begin{bmatrix} x\\ y \end{bmatrix}$, 写出 $\sigma(\alpha)$ 的具体表达式。
\item  证明 $\sigma$ 是线性变换，并写出 $\sigma$ 关于标准基 $\{\varepsilon_1, \varepsilon_2\}$ 的矩阵。
\item  验证 $\langle \sigma(\alpha), \sigma(\alpha) \rangle = \langle \alpha,\alpha \rangle$, 从而证明 $\sigma$ 是正交变换。
\end{enumerate}

%\begin{center}
\includegraphics[height=4cm,width=6cm]{exam-8-3-a.png}
%\end{center}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %2
设 $V=\mathbb{R}^2$ 是平面，设 $\sigma: V\to V$ 由 
$\sigma \left( \begin{bmatrix} x\\ y \end{bmatrix} \right) 
= \begin{bmatrix} 2&2 \\ 2&5 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} $ 定义。
\begin{enumerate}
\item  按定义验证 $\sigma$ 是对称变换。
\item  将矩阵 $A=\begin{bmatrix} 2&2 \\ 2&5 \end{bmatrix}$ 正交相似于对角矩阵。
\item  求规范正交基 $\{\gamma_1, \gamma_2\}$ 使得 $\sigma$ 关于这个基的矩阵是对角矩阵。
\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{enumerate}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}





